Inspired from the constructive method of Davilla et al. 10, with new ingredients, we extend their existence results to dimensions 7≤n≤9 concerning the following Hénon type ...problem{−Δu=K|u|p−1−εuinΩ,u=0on∂Ω, where Ω is a smooth bounded domain in Rn, ε is a positive real parameter, p+1=2n/(n−2) is the critical Sobolev exponent and the function K∈C2(Ω‾) is positive satisfying condition (1.1).
In this note, we obtain a classification result for positive solutions to the critical p-Laplace equation in Rn ${\mathbb{R}}^{n}$ with n ≥ 4 and p > p n for some number pn∈n3,n+13 ${p}_{n}\in ...\left(\frac{n}{3},\frac{n+1}{3}\right)$ such that pn∼n3+1n ${p}_{n}\sim \frac{n}{3}+\frac{1}{n}$ , which improves upon a similar result obtained by Ou (“On the classification of entire solutions to the critical p-Laplace equation,” 2022, arXiv:2210.05141) under the condition p≥n+13 $p\ge \frac{n+1}{3}$ .
We consider the problem −div(α(x)|∇u|p−2∇u)=λ|u|q−2u+|u|p⋆−2u in a bounded domain, with homogeneous Dirichlet boundary condition, where α(.) is a continuous function, p⁎ the critical Sobolev exponent ...and 2≤p≤q<p⁎. We prove some existence of positive solutions which depends, among others, on the behavior of the potential α(.) near its minima, the position of p2 with respect to the dimension of the space and on the position of q with respect to some precise values.
We investigate the blow-up behavior of sequences of sign-changing solutions for the Yamabe equation on a Riemannian manifold (M,g) of positive Yamabe type. For each dimension n≥11, we describe the ...value of the minimal energy threshold at which blow-up occurs. In dimensions 11≤n≤24, where the set of positive solutions is known to be compact, we show that the set of sign-changing solutions is not compact and that blow-up already occurs at the lowest possible energy level. We prove this result by constructing a smooth, non-locally conformally flat metric on space forms Sn/Γ, Γ≠{1}, whose Yamabe equation admits a family of sign-changing blowing-up solutions. As a counterpart of this result, we also prove a sharp compactness result for sign-changing solutions at the lowest energy level, in small dimensions or under strong geometric assumptions.
This paper deals with the following Kirchhoff type problem (0.1)−a+b∫RN|∇u|2Δu=λK(x)uq−1+u2∗−1,u>0,x∈RN,where N≥3, K∈L1(RN)∩L∞(RN), constants a,b>0, the parameter λ>0 and 2≤q<2∗≔2N∕(N−2). Using the ...finite-dimensional reduction approach, we prove that problem (0.1) admits at least a positive solution.
In this work, by using variational methods we study the existence of
nontrivial positive solutions for a class of p-Kirchhoff type problems with
critical Sobolev exponent.
In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated ...with critical Sobolev exponent in $\mathbb{R}^N$. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the $L^2$ norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900.
We study the perturbation by a critical term and a (p - 1)-superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments ...and a version of the concentration-compactness principle in the singular case, we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem.
PDF
